Data in health-related sciences are often measured on categorical scales. Tables that display such data are often sparse, having few observations in many categories. Two common reason are (1) the study may have a small number of subjects, because of cost, ethics, or patient availability, or (2) repeated measurement of responses may produce a large multidimensional table. The proposed research focuses on developing statistical methodology for such sparse data. Small-sample analyses: Exact and nearly exact methods will be developed for making inferences about associations between categorical responses. New methods include a modified test of conditional independence that is less conservative than the usual one; a corresponding confidence interval for an odds ratio effect in the stratified 2X2 case in narrower than the usual one. Exact statistical inference will be further developed for associations and interactions involving ordered categorical responses. Repeated categorical measurement data: Methods will be further developed to describe how categorical response vary across occasions and according to values of covariates, with special emphasis on the ordinal-response case. Models to be developed include parametric and non-parametric versions of models for subject-specific effects, latent class and mixture models, and models for marginal distributions. A newly developed maximum likelihood algorithm will be utilized to fit some models that are awkward to handle with standard methods. Models will be applied to relevant biomedical problems. For instance, exact methods are useful for making comparison of treatment on a categorical response when the sample size is small and large-sample approximations are untrustworthy. The subject-specific models are useful for subjects-wise comparisons of treatments in cross-over studies and in describing occasion effects in randomized clinical trials that involve making observations at the beginning and end of a treatment period. Latent class models are useful assessing inter-rater reliability.